"Some forms of representation -- such as diagrams, graphical displays, and symbolic expressions -- have long been part of school mathematics. Unfortunately, these representations and others have often been taught and learned as if they were ends in themselves. Representations should be treated as essential elements in supporting students' understanding of mathematical concepts and relationships."

(NCTM, 2000, p. 67)

When we solve mathematical problems, a core part of the solution process is how we represent the ideas in the problem. The form of representation we select allows us to manipulate the information (as opposed to manipulating symbols) to reach a sensible solution. With the Representation Standard, it is our goal for students to learn to recognize, compare, and use an array of representational forms for the concepts they are learning in the middle grades.

Representation is not about manipulating symbols. Rather, it requires students to generate or select a meaningful way to show the facts in a mathematical situation. Using representation helps them identify patterns and relationships and make predictions.

Students in the middle grades should use multiple representations, choosing the appropriate representation for a particular problem situation. Examples of forms of representation include models, drawings, tables, graphs, spreadsheets, expressions, and equations. When students gain access to mathematical representations and the ideas they represent, they have a set of tools that enable them to organize their thinking and increase their ability to think mathematically.

It is not usually apparent which form of representation to use when students first approach a problem. They might find that their own representation is the most comfortable with which to begin. Part of problem solving is trying, and sometimes rejecting, possible forms of representation when choosing an effective one to use.